Squeeze Theorem

The theorem, stated precisely

Suppose g, f, and h are defined on a punctured neighbourhood of a, and

g(x) ≤ f(x) ≤ h(x)  for all x near a (x ≠ a),
lim_{x→a} g(x) = lim_{x→a} h(x) = L.

Then lim_x→a f(x) = L. The result extends without change to x → ±∞, one-sided limits, and to sequences (replace the punctured neighbourhood by "for all sufficiently large n").

The proof is two lines of ε-δ. Given ε > 0, pick δ small enough that |g(x) − L| < ε and |h(x) − L| < ε whenever 0 < |x − a| < δ. Then L − ε < g(x) ≤ f(x) ≤ h(x) < L + ε, so |f(x) − L| < ε. Done. No differentiability, no continuity, no smoothness — only the inequality and the two outer limits.

How to apply it

Given a tricky limit lim_x→a f(x), the squeeze recipe is:

1. Find bounds g(x) ≤ f(x) ≤ h(x) on a neighbourhood of a.
2. Compute lim_x→a g(x) and lim_x→a h(x) — usually by direct substitution.
3. If both limits equal the same L, then lim f = L.
4. If the limits differ, the squeeze fails — find tighter bounds, or use a different method.

The craft is step 1. Tight enough bounds let you conclude, loose bounds leave you stuck. The most common pattern is "bounded × infinitesimal": if |f(x)| ≤ M is bounded and ε(x) → 0, then f(x) · ε(x) is sandwiched between −Mε(x) and Mε(x), both of which go to 0.

Worked example — proving lim_x→0 sin(x)/x = 1

This is the canonical squeeze-theorem application and the foundation of derivatives for trig functions. The naive bound −1 ≤ sin(x)/x ≤ 1 doesn't squeeze — both bounds are constants disagreeing at the limit. We need a geometric bound from the unit circle.

For 0 < x < π/2, draw a unit circle, a sector of angle x, the inscribed triangle (vertices at the origin, (1, 0), and (cos x, sin x)), and the outside triangle with vertices at the origin, (1, 0), and (1, tan x). Comparing areas:

area(inner triangle)  =  (1/2) sin(x)
area(sector)          =  (1/2) x
area(outer triangle)  =  (1/2) tan(x)

⇒  sin(x) ≤ x ≤ tan(x)  for 0 < x < π/2.

Divide through by sin(x) (positive for the range), then take reciprocals (flips inequalities):

1 ≤ x/sin(x) ≤ 1/cos(x)
cos(x) ≤ sin(x)/x ≤ 1

As x → 0⁺, cos(x) → 1 (direct substitution; cos is continuous). The outer bounds converge to 1. By the squeeze theorem, sin(x)/x → 1. The left-handed limit follows by symmetry (sin(−x)/(−x) = sin(x)/x is even). The geometric squeeze is honest — every step comes from areas on the unit circle.

Worked example — x²sin(1/x) → 0 as x → 0

The factor sin(1/x) oscillates infinitely fast as x approaches 0 — between −1 and 1, infinitely often, on every neighbourhood of 0. L'Hôpital won't help: the derivative ratio explodes. Direct substitution gives 0 · undefined. But the squeeze is trivial:

For x ≠ 0:  |sin(1/x)| ≤ 1
            −x² ≤ x²·sin(1/x) ≤ x²

lim_{x→0} (−x²) = 0,  lim_{x→0} x² = 0.

By squeeze, x² · sin(1/x) → 0. The infinite oscillation gets absorbed by the x² factor that's crushing everything to zero. This is the cleanest illustration of the "bounded × infinitesimal" pattern.

The same idea extends to x · sin(1/x) — bounded between −|x| and |x|, hence → 0. But it does not extend to sin(1/x) alone — that has no limit at 0, because the oscillation doesn't dampen.

Variants and stronger theorems

• One-sided squeeze. If the inequality holds only for x > a (or x < a), the conclusion still holds for the corresponding one-sided limit. Useful when functions are defined piecewise around a.
• Sequence version. aₙ ≤ bₙ ≤ cₙ for n ≥ N, aₙ → L, cₙ → L ⇒ bₙ → L. The proof rewrites δ as N; identical idea.
• Infinite limits. If g(x) ≤ f(x) and g(x) → ∞, then f(x) → ∞ — a one-sided version with only a lower bound. Symmetrically for −∞ with an upper bound.
• Squeeze for continuity. If g, f, h are continuous at a, g(a) = h(a) = L, and g ≤ f ≤ h locally, then f is continuous at a with f(a) = L. Often used to define f at a by limiting argument.
• Multivariable squeeze. Same statement in ℝⁿ with the neighbourhood being a punctured ball. Bounds usually involve ||x − a|| or polar coordinates.

When the squeeze is the right tool

Situation	Use squeeze?	Why
Bounded oscillation × infinitesimal	Yes — first choice	The bound −M·ε ≤ f ≤ M·ε is immediate
0/0 with simple derivatives	No — L'Hôpital	Differentiating is faster and mechanical
Smooth function with no closed-form bound	Often no	Taylor expansion gives leading-order behaviour directly
Sequence with known error bound	Yes	If |aₙ − L| ≤ εₙ → 0, squeeze gives aₙ → L
Multivariable limit in polar form	Yes — natural fit	||x − a||ᵏ bounds work cleanly in polar
Limit doesn't exist (oscillates)	No	No squeeze possible; argue directly via sequences

Common mistakes

• Bounds that don't converge to the same L. If lim g ≠ lim h, the squeeze tells you nothing. Loose bounds are useless; tighten them.
• Forgetting "near a." The inequality only needs to hold on a punctured neighbourhood — values at a itself don't matter, and you can usually ignore them.
• Flipping inequalities without flipping signs. Multiplying by a negative number reverses the direction. Same for taking reciprocals of negatives.
• Confusing squeeze with L'Hôpital. L'Hôpital differentiates the original ratio; squeeze finds outer bounds. They solve different problem shapes.
• Squeezing the wrong thing. For lim x·sin(x), squeeze fails because x · sin(x) is bounded between −|x| and |x| — that gives → 0, which is wrong if you confuse x → 0 with x → ∞. Always check the limit point matches the bounds.
• Using squeeze when bounds don't exist. sin(1/x) has no limit at 0 — no monotone bound shrinks to a single L. Squeeze is silent; the limit simply doesn't exist.

Connections to other techniques

L'Hôpital's rule. Squeeze handles oscillating bounded factors that L'Hôpital cannot — sin(1/x), cos(1/x), Dirichlet-like indicator functions. Conversely, L'Hôpital wins on smooth 0/0 forms where the derivatives are simple polynomials or exponentials.

Taylor expansion. A Taylor remainder bound |Rₙ(x)| ≤ M|x − a|ⁿ⁺¹/(n+1)! is exactly a squeeze. Truncating after n terms gives an error that is squeezed to zero, justifying Taylor approximations.

Numerical analysis. Convergence proofs for iterative algorithms typically squeeze the iteration error between geometric or polynomial decay rates and zero. The convergence rate is the squeeze rate.

Real analysis foundations. Squeeze is one of the first theorems you prove after the ε-δ definition of limit, because it lets you transfer convergence from outer functions to inner ones without computing the inner limit directly.